Les Cahiers du CREF
ISSN: 1707-410X
Swap Rate Movements via the Target Rate: a No-arbitrage Regime-Switch Approach René Ferland Geneviève Gauthier Simon Lalancette CREF 08-03 April 2008
Copyright 2008. René Ferland, Geneviève Gauthier, Simon Lalancette. Tous droits réservés pour tous les pays. Toute traduction et toute reproduction sous quelque forme que ce soit sont interdites. Les textes publiés dans la série «Les Cahiers du CREF» de HEC Montréal n'engagent que la responsabilité de leurs auteurs. La publication de cette série de rapports de recherche bénéficie d'une subvention du programme de l'Initiative de la nouvelle économie (INE) du Conseil de recherches en sciences humaines du Canada (CRSH).
Swap Rate Movements via the Target Rate: a No-arbitrage Regime-Switch Approach
René Ferland Department of Mathematics Université du Québec à Montréal
[email protected] Geneviève Gauthier Department of Management Sciences HEC Montréal and CREF
[email protected] Simon Lalancette Department of Finance HEC Montréal
[email protected] HEC Montréal 3000, chemin de la Côte Sainte-Catherine Montréal (Québec) H3T 2A7 CANADA
April 2008
Les Cahiers du CREF CREF 08-03
Les Cahiers du CREF
CREF 08-03
Swap Rate Movements via the Target Rate: a No-arbitrage Regime-Switch Approach Abstract This study examines the impact of target rate movements on the interest rate swap curve in a no-arbitrage environment. The model rests upon two observable state variables, the target and the Fed Funds rates, whose dynamics are embodied into a three-state regime-switch environment associated with FOMC monetary actions. Risk-neutral pricing of zero-coupon bonds is based on an approximate recursive form whose empirical implementation demands much less than the Monte Carlo valuation. The model implies that the market price of risk consists as an affine function of the state variables with regime-dependant sensitivity coefficients. Given the observability of the state variables, empirical implementation is achieved using a two-step approach. The findings support the regime-switch orientation of the model which captures successfully the joint dynamics of the target and Fed Funds rates. Successive calibrations result, in general, in very small pricing errors. A sensitivity analysis based on the model shows that transitions between regimes can drastically modify the shape of the interest rate swap curve. Keywords:
Regime-Switch, Target rate, Swap curve, valuation, FOMC regime, Hidden Markov chain.
Risk-neutral
JEL Classification: G1, C1
Acknowledgements: The authors acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) (Ferland and Gauthier) and by HydroQuébec (Lalancette).
Les Cahiers du CREF
CREF 08-03
Résumé Cet article étudie l’impact des mouvements du taux cible sur les mouvements de la structure à terme des taux d’intérêt swap dans un environnement caractérisé par l’absence d’arbitrage. Le modèle s’appui sur deux variables d’états, soit le taux cible et le taux « Fed Funds ». Les fluctuations de ces variables sont régies par une chaîne de Markov comportant trois états associées aux actions monétaires du FOMC. La tarification des obligations zéro-coupons découle d’une forme récursive approximative, s’inscrivant dans un cadre risque-neutre, substantiellement moins exigeante que l’approche Monte Carlo. La prime de risque dépend de la dynamique des variables d’états via une spécification affine où les coefficients de sensibilité dépendent du régime sous-jacents aux actions monétaires. L’estimation empirique du modèle s’effectue en deux étapes grâce à l’observation des variables d’états. Il apparaît que la technologie de changements de régimes capte adéquatement le comportement du taux cible et du taux « Fed Funds ». Par ailleurs, les erreurs de tarification dégagées par une calibration successive, sont très petites. Finalement, le modèle montre que les changements de régimes provenant des actions du FOMC, peuvent modifier drastiquement la forme de la structure à terme des taux d’intérêt swap Mots clés : Changements de régime, Taux cible, Courbe swap, Tarification risque-neutre, Régime du FOMC, Chaîne de Markov.
Swap Rate Movements via the Target Rate: a No-arbitrage Regime-Switch Approach René Ferland, Geneviève Gauthieryand Simon Lalancettez, x First draft: March 2008, this draft: April 18, 2008.
Abstract This study examines the impact of target rate movements on the interest rate swap curve in a no-arbitrage environment. The model rests upon two observable state variables, the target and the Fed Funds rates, whose dynamics are embodied into a three-state regime-switch environment associated with FOMC monetary actions. Riskneutral pricing of zero-coupon bonds is based on an approximate recursive form whose empirical implementation demands much less than the Monte Carlo valuation. The model implies that the market price of risk consists as an a¢ ne function of the state variables with regime-dependant sensitivity coe¢ cients. Given the observability of the state variables, empirical implementation is achieved using a two-step approach. The …ndings support the regime-switch orientation of the model which captures successfully the joint dynamics of the target and Fed Funds rates. Successive calibrations result, in general, in very small pricing errors. A sensitivity analysis based on the model shows that transitions between regimes can drastically modify the shape of the interest rate swap curve. Keywords: Regime-Switch, Target rate, Swap curve, Risk-neutral valuation, FOMC regime, Hidden Markov chain.
1
Introduction
Movements in interest rate constitute a central preoccupation within the …nancial community. Changes in Treasury and swap curves not only convey a substantial amount of information but also impact the pricing of …xed income instruments and related interest rate derivatives. Of primary importance, therefore, is the reaction of yield curves to information releases and to market movers such as the target rate of the Federal Reserve (FR). FOMC Associate Professor, Department of Mathematics, University of Quebec in Montreal. PO Box 8888, Downtown Station, Montréal, H3C 3P8.
[email protected] y Associate Professor, Department of Quantitative Methods, HEC-Montréal. 3000 Chemin de la CoteSte-Catherine, Montreal, H3T 2A7.
[email protected] z Professor, Finance Department, HEC-Montréal.
[email protected] x Research supported in part by NSERC grant (Ferland and Gauthier) and by Hydro-Québec (Lalancette).
1
meetings attract substantial coverage and movements in the target rate in‡uence interest rate swaps and options. But the precise mechanisms by which this causality takes place do not appear being fully understood. Thus, as a step towards this direction, this study aims at investigating the dynamic reaction of the US interest rate swap curve in response to movements in the FR target rate. The staggering amount of research that has taken place in the area of dynamic term structure modeling has established the absence arbitrage as the dominant methodological paradigm in yield curve modeling ([Dai and Singleton (2003)]). The imposition of the noarbitrage constraint suggests that movements in the Federal Funds (FF) rate, as a response to monetary actions undertaken by the FR, have repercussions over the entire yield curve. [Piazzessi (2005)] targets a similar goal and resorts to an approximately a¢ ne framework ([Du¢ e and Kan (1996)]) when the target rate jumps on FOMC meetings with jump intensities conditioned upon the state variables. This paper focuses on a similar issue but adopts a di¤erent path of investigation. It is typical practice in this area of research to model the dynamics of latent state variables. But the inevitable presence of measurement errors involved with this approach may induce a low correlation amongst, for instance, the extracted state variables and the bond prices (see [Piazzessi (2005)], amongst others, for a related discussion). Matching the entire yield curve in this context can become a di¢ cult task. Therefore, in contrast to this tendency, the approach undertaken herein is articulated over observable state variables, namely the target rate and its companion rate, the FF rate as a proxy for the short rate. At least two favorable features emanate from this orientation. First, observability of that factor ensures internal consistency in the joint modelization of target rate and its corresponding short rate. Second, the FF rate is strongly correlated with the swap rates at least for the sample used herein1 . This should help the model to match the long end of the yield curve2 . As in [Dai, Singleton and Yang (2007)] and [Bansal and Zhou (2002)], the factor model devised herein is based on a regime-switching technology ([Hamilton (1988)]). In contrast to [Piazzessi (2005)] who resorts to a dual jump process to capture the dynamics of the target 1
Over the sample period described below, the correlations between the FF rate and the 2, 5, 7 and 10 year swap rates are respectively 0.93, 0.83, 0.78 and 0.71. 2 Concerns over the time series properties of the FF rate are raised by [Hamilton (1996)] and by [Piazzessi (2003)]. They advocate the presence of seasonal measurement errors in the FF rate series as a response to the so-called "settlement Wednesday" e¤ect. Until July 1998, the reserve computation period imposed to banks, was overlapping with the reserve maintenance period which would end on a Wednesday. Uncertainty surrounding the amount of required reserves, would resolve only on the last Wednesday of the computation/maintenance period, creating seasonal spikes in the FF rate. However, the FR introduced on July 1998, a lagged computation period which eliminates the overlap. [Kotomin and Winters (2007)] mention that the FF rate demonstrates a much smoother behavior after July, 1998 and contend that the "settlement Wednesday" e¤ect has vanished.
2
rate, a time-heterogeneous Markov chain is used. One of the advantage of this approach, is the avoidance of possible negative jump intensities. It is assumed that the target rate moves eight times a year, at equal time intervals, following an FOMC meeting. At each meeting, the transition probability matrix driving the dynamics of the target rate is conditioned upon a hidden three-state Markov process. In the …rst (second) "FOMC" regime, only probabilistic upward (downward) movements of the target rate are considered. In the last regime, labeled as "stable", target rate movements can take either direction. Overall, the target rate invariably moves by multiple of 25 basis points but the transition probability matrices di¤er across the regimes. As the FF rate must revert back to the target rate, it seems natural in this framework to con…ne its speed of target-reversion and volatility to a regime-switch framework in association with the FOMC policy regimes. Because of the Markov chain, a closed-form solution for zero-coupon bond pricing is numerically impracticable since the number of terms increases exponentially with the maturity. Thus, an analytical approximation, in which there is no risk premium for regime transitions, is derived and compared to a Monte Carlo valuation. In the model, the market price of risk is an a¢ ne function of the state variables whose coe¢ cients are regime-dependant. On the empirical side, the presence of two Markov chains leads to a proliferation of parameters. The estimation is thus disentangled into a two-step procedure3 . In the …rst step, the set of parameters that characterize the dynamics of the state variables are estimated from the time series of the target and FF rates. In the second step, the risk-neutral parameters are obtained by means of calibration. The remainder of the paper is organized as follows. In the next section, a short literature review is presented. In section 3, the data are described. Section 4 presents the model while section 5 discussed the estimation and calibration steps. The empirical …ndings are discussed in Section 6. Finally, concluding remarks are o¤ered in the last section.
2
Related literature
A large portion of the literature devoted to dynamic yield curve modeling, focuses on generalized a¢ ne processes usually inspired from [Vasicek (1977)] and [Cox, Ingersoll and Ross (1985)]. This trend has largely bene…ted from the seminal work of [Du¢ e and Kan (1996)] who derive the set of ode that characterize the time series behavior of the no-arbitrage coe¢ cients of the state variables that enter into the pricing of zero-coupon bonds. Further improvements on this type of models are o¤ered by [Dai and Singleton (2000)]. They identify the set of restrictions imposed on the volatility parameters of generalized square root processes, 3
See, for instance, [Ang and Piazzesi (2004)] and [Chun (2007)].
3
in order to preserve their admissibility to the a¢ ne class. In this category of models, the market price of risk is an a¢ ne representation of the state variables. E¤orts to come up with more ‡exible representations of the market price of risk are found in [Du¤ee (2002)] and [Duarte, J., (2003)]. [Piazzessi (2003)] provides a survey on a¢ ne term structure models. A richer speci…cation of the state variables is obtained with quadratic a¢ ne models (see for example, [Anh, Dittmar and Gallant (2002)] and [Ang, Boivin and Dong (2007)]). The parameterization gain is not costless, however, since the exclusive mapping between yields and latent factors vanishes in addition to a loss of parsimony ([Dai and Singleton (2003)] provide a critical survey of these models). Additional comprehension of the dynamic properties of bond yields is gained by including jumps into a¢ ne or quadratic speci…cations. However, such a technology cannot account for the occurrence of structural changes in bond yield distributions as a response, for example, to di¤erent monetary regimes pursued by the FR. The regime-switch approach is more suited for that task. While it has been widely applied to a variety of empirical issues, incorporation of regime-switch into dynamic yield curve modeling, is fairly recent. [Bansal and Zhou (2002)] derive and estimate a two-factor CIR speci…cation embodied into a two-state regime-dependant risk-neutral pricing framework. They conclude on the superiority of their regime-switch term structure model against a standard three factor CIR model. [Dai, Singleton and Yang (2007)] and [Ang, Bekaert and Wei (2008)] propose more sophisticated versions of term structure models with regime-switch. The former authors device a three-factor (correlated) Gaussian structure with mean reversion and volatility parameters that are regime-switch. The model also includes a risk premium attributed to regime-shift risk. As expected, these characteristics allow the model to better capture the key features of excess bond returns. Using observable time series as state variables obviously reduces the ambiguity surrounding the nature of the pervasive forces at work in term structure applications. Otherwise, model misspeci…cations may result in extracted latent variables that diverge signi…cantly from their label (see [Piazzessi (2005)] and [Bikbov and Chernov (2005)] for an interesting discussion). E¤orts to consider macro-factors as state variables in term structure modeling are found in [Ang and Piazzesi (2004)] (combination of observable and latent factors) and [Chun (2007)] (observable factors only), amongst other. Two-step estimation procedures are implemented in this case.
4
Figure 1: Federal Funds and Target Rates Over the Sample Period
3
Data
Two sets of data are required for this study. The …rst (second) data set relates to the statevariables (swap rates). The target rate series and its companion rate, the e¤ective FF rate, are obtained from the Board of Governors of the FR website4 . Following the conclusions of [Kotomin and Winters (2007)] with respect to the properties of the FF rate, the observation period begins on the …rst week of August 1998 and ends on the last week of December 2007. Yield data are represented by the US interest swap curve. This market is very liquid5 , and the transaction process is fairly standardized with very frequent quotations of rates with constant maturities6 . For many years now, the swap curve has been considered as an alternative benchmark to Treasuries securities. Therefore, for every quarter of the sample period, starting on the second week of August 1998 and ending on the last week of December 2007, the swap rates with maturities of 2, 5, 7 and 10 years are extracted from Bloomberg. FOMC meetings observed during the sample period, are mostly held on Tuesday, but sometimes on Wednesdays or Thursday, usually for one day and occasionally for two days. Nevertheless, on meeting weeks, monetary actions dictated by the FOMC, are always known by Thursday. Therefore, FF and swap interest rates are all extracted on the Fridays of each week. Figure 1 shows the target and FF rate series for the sample period. As expected, it is observed that the FF rate quickly reverts to the target rate which ‡uctuates within upward and downward periods separated by periods of stability. 4
www.federalreserve.gov According to [BIS (2007)], US interest rate swaps outstanding in 2007 amount to 1; 114; 311 billions of dollars. 6 Given the analysis of [Du¢ e and Singleton (1997)], we conform to the usual practice where newly issued interest swaps are considered as par bonds discounted at the collection of appropriate Libor rates. 5
5
4
Modeling Approach
Let
t
and
t
denote the unobserved regime of the FR at time t and its target rate, respec-
tively. Figure 1, which displays the target and its companion rate over the sample period, suggests that a regime switching model based on three regimes seems warranted. These dynamics are modeled by the …rst four assumptions. Assumption 1 FOMC meetings are held eight times a year. The meeting dates are known and equally spaced during a year, that is tk = k
t,
where
t
=
1 8
is the time (in year) elapsed
between two consecutive meetings. This time frame is too coarse for the modeling purposes. The mismatch between the frequency at which the FOMC meetings are held and the sampling frequency of the swap curves is easily handled by assuming that f( t ; t ) : t 2 ft0 ; t1 ; t2 ; :::gg
behaves as a time-inhomogeneous Markov chain. For those time periods without a FOMC meeting, the transition probability matrix is just the identity matrix; otherwise, it takes the form: i h 0 0 0 0 P ( ; ); ( ; ) = P tn+1 = ; tn+1 = tn = ; tn = i h i h 0 0 0 = ; = = P tn+1 = = ; = P = = ; t t t n n tn+1 tn tn n+1 h i h i 0 0 = P tn+1 = P tn+1 = tn = ; tn = tn = = M ( ; 0 )M ( ; 0 ) :
(1)
Notice that the third equality implies that the transition probabilities of the Markov chains associated with FOMC regimes are not a¤ected by the level of the target rate. This reduces the number of parameters involved in the estimation. Assumption 2 The dynamics f( t ; t ) : t 2 ft0 ; t1 ; t2 ; :::gg of the target rate and the FOMC regime are the same under the Q and P measures.
The next two assumptions de…ne the transition probability matrices of the Markov chains. Assumption 3 The FR policy regimes implies a transition matrix that corresponds to 0 1 pdd pds pdu M = @ psd pss psu A pud pus puu
on FOMC meetings dates and to the identity matrix otherwise.
6
Assumption 4 The target rate,
behaves as a Markov chain with a transition probability
t;
matrix that depends upon the FOMC policy regime at each meeting date. Given an upward, downward or a stable FOMC regime, t has a corresponding probability matrix labeled Mu , Md or Ms . In addition,
t
moves by multiples of 25 basis points but is bounded to 50 basis
7
points changes . Furthermore, entries in the matrices Mu and Md are such that the target rate cannot decrease (increase) in the upward (downward) FOMC regime. Assumption 5 The dynamic of the short rate follows a target-reverting process under both data-generating measure P or risk-neutral measure Q : drt =
P
drt =
Q
tn
(
tn
rt ) dt +
tn
dWtP ;
(2)
(
tn
rt ) dt +
tn
dWtQ ;
(3)
and with tn < t
tn
tn+1 and where W P (W Q ) is a P
The parameters
P u,
P s,
P d
Q u,
(
Q d)
Q s,
(Q ) Brownian motion.
capture the speed of target-reversion across the
di¤erent FOMC regimes under the P (Q) measure. Similarly, u , s and d represent the volatility parameters under each FOMC regime. As the FF rate, which proxies the short rate, drifts very closely to the target rate as shown in Figure 1, it is anticipated that the reversion (volatility) parameters will display large (small) values. Rt Replacing WtP by WtQ ds in Equation (2) where f t : t 0 s gives drt =
P
tn
(
rt )
tn
tn
t
dt +
tn
0g is the risk premium,
dWtQ :
A direct comparison of the drift terms of the latter SDE and Equation (3) gives P t
Q tn
tn
=
(
rt ) ; tn < t
tn
(4)
tn+1 :
tn
This last equation demonstrates that the market price of risk is an a¢ ne representation of the state variables. Also changes in FOMC regimes drive the market price of risk through the reversion and volatility parameters of rt . In the following, some notation is required. Let fFt : t
generated by the observable variables
0g) be the …ltration
and r ( ; r and ). That is
Ft =
f s ; ru : s 2 ft0 ; t1 ; t2 ; :::g with ts
Gt =
f s ; ru ;
s
0g (fGt : t
: s 2 ft0 ; t1 ; t2 ; :::g with ts
7
t and 0 t and 0
u
tg ; u
tg :
Although more important movements in the target rate have been observed in the history (consider, for instance, the substantial rate decreases observed during the 2008 winter), no target rate change beyond 50 bps are found in the sample used in this study.
7
As usual, the time t value of a zero-coupon bond paying one dollar at time T is Z T Q ru du Ft : P (t; T ) = P (t; T ; rt ; t ) = E exp
(5)
t
Under the framework presented above, the solution to equation (5) exists but involves a bewildering number of exponential terms that grow exponentially with T . This feature renders the empirical implementation impractical. Inspired from [Bansal and Zhou (2002)], a recursive approximation solution is derived in the next theorem given the observability of the initial regime. Theorem 6 Let t1 < t2 < ::: < tn < ::: be the time discretization where tn = n t
= kmax
t.
Assuming that the current short rate rtn , the target rate
tn
t
with
and its current
regime tn are all observed, the time tn value of the zero-coupon bond paying one dollar at time tn+m may be approximated by P tn ; tn+m ; rtn ;
tn ;
tn ; kn
tn ; kn
= exp Am
rtn + Bm
tn ;
(6)
tn ; kn
where kn 2 f0; 1; :::; kmax g corresponds to the number of time periods since the last meeting, and the coe¢ cients A and B are determined recursively. If 1
e
2
Q
t
= 1
e
Q
t
and
=
then 2
2
A1 ( ; k) =
, B1 ( ; ; k) = Q
+
t
Q
4
Q 3
Q 2
t Q
2
(7)
and, for m 2 f2; 3; :::g, Am ( ; k)
Bm ( ; ; k)
=
=
(
Q
+ Am
Q
+ Am
1
( ; k + 1) e
1
( ; 0) e
t t
if k 2 f0; 1; :::; kmax
1g
if k = kmax
8 " # > Am 1 ( ; k + 1) + Q + B m 1 ( ; ; k + 1) > t > > if k 2 f0; 1; :::; kmax > > > + 12 vR ( ) + 21 A ( ; k + 1) vr ( ) Am 1 ( ; k + 1) cr;R ( ) < " > > > > > > > :
Am
1
( ; 0)
+
Q
+ 21 vR ( ) + 21 A ( ; 0) vr ( )
t
+ Bm
Am
8
1
1
( ; ; 0)
( ; 0) cr;R ( )
#
if k = kmax
1g
where Am
1
( ) =
Am
1
( ) =
Bm
1
( ; ) =
X X
Am
1
( )Q
tn+1
=
tn
=
;
A2m
1
( )Q
tn+1
=
tn
=
;
Bm
1
( ;
X
)Q
tn+1
=
;
tn+1
=
tn
= ;
tn
=
;
;
2
vR ( ) =
t
Q 2
2
Q
+
1 2
Q
;
2
vr ( ) =
Q
2
;
2
cr;R ( ) =
Q 2
2
:
The proof is given in Appendix A. Since the regime of the target rate is not an observable variable, the time tn value of the zero-coupon bond is approximated by X P (tn ; tn+m ; kn ) = P tn ; tn+m ; rtn ; tn ; tn ; kn Q tn =
:
(8)
If no information about the current regime of the target rate is available, the probabilities Q
tn
=
may be set to the stationary distribution of the Markov chain or, otherwise the
…ltered probabilities may be used. If there are n = 3 regimes and n states for the target rate, then for each m, n kmax values for Am and n n kmax values for Bm are required. To assess the quality of the Approximation (8), a Monte Carlo study is performed when the parameters have been chosen randomly according to the distributions presented in Table 1. The analytical approximation is constructed on a weekly basis with
t
=
1 48
and kmax = 6,
i.e. six time periods between two consecutive FOMC meetings. A thousand sets of parameters are simulated. For each of them, the zero-coupon bond prices and related yields inherent to the analytical approximation and to the Monte Carlo valuation based on 10 000 runs, are computed. Table 2 reports the …nding for a maximum maturity of 10 years8 . Overall, the …ndings suggest that the analytical approximation performs well as 50% of the pricing errors are smaller than 9:1 basis points in the 10 year case. For shorter maturities, the quality of the approximation further improves9 . Some caution 8
The Appendix A shows that the passage from line (12) to line (13) can be obtained directly using Jensen’s inequality. Therefore, P tn ; tn+m ; rtn ; tn ; tn exp Am tn rtn + Bm tn ; tn such that the analytical approximation underestimates the true bond price. Furthermore, since the Jensen’s inequality is used recursively, the pricing error grows with the time to maturity. 9 Notice that a few cases generated negative pricing errors. These anomalous …ndings occur because of the imprecision of the Monte Carlo benchmarks in these cases.
9
Table 1: Parameter’s Distribution of the Monte Carlo Study Parameters
Distribution
Model’s constraints
Regime’s transition probabilities pdd ; pss , puu U (0:7; 1) pdu = 0 pud = 0 psd U (0; 1 pss )
pds = 1 pus = 1 psu = 1
Target rate’s transition probabilities pd (0) ; ps (0) ; pu (0) U (0; 1) pu (25) U (0; 1 pu (0)) pd (25) U (0; 1 pd (0)) ps ( 50) U (0; 1 ps (0)) ps ( 25) U (0; 1 ps (0) ps ( 50)) ps (25) U (0; 1 ps (0) ps ( 50) ps ( 25)) Spot rate’s parameters d; s; u d; s; u
pdd puu pss
psd
pu (50) = 1 pd (50) = 1
pu (25) pd (25)
ps (50) = 1
ps (0)
pu (0) pd (0)
ps ( 50)
ps ( 25)
ps (25)
U (0; 100) U (0:005; 0:045)
Initial values Uniformly distributed over the target rate’s state space:f0:005; 0:0075; 0:0100; ..., 0:0700g
0
r0 =
0
U (a; b) denotes the uniform distribution over the interval [a; b]. p (k) corresponds to the conditional probability, under the actual regime , that the target rate moves by k basis points. It is assumed that switches from the upward to the downward FOMC regimes or, vice-versa, must transit by the stable regime. Thus, pdu = pud = 0. The model’s constraints require that the rows of any transition matrix must sum to one.
10
Table 2: Descriptive Statistics of the Pricing Errors for the 1, 3, 7 and 10 year Zero-coupon Yields (in basis points) Time to maturity
1
Mean Median Standard deviation Min Max
3
7
10
0,5930 4,1481 10,5922 13,7937 0,4790 2,9765 7,2710 9,1080 0,6612 3,9946 11,4679 15,9270 -1,0540 -0,9540 -0,2450 -0,0700 3,9650 29,6830 76,7750 109,8100
The reported statistics are based on a sample of size 1000. The errors are computed as the di¤erence between the analytical approximation and the corresponding Monte Carlo yields. The Monte Carlo estimates are generated based on 10 000 sample paths.
Figure 2: Pricing Errors, Expressed in Basis Points, for the 10-year Zero-Coupon Bond Yield Distribution's error of the ten years zero-coupon yields (basis points) 600
500
400
300
200
100
0 -20
0
20
40
60
80
100
120
The errors are computed as the di¤erence between the analytical approximation and the corresponding Monte Carlo yields with a maturity of 10 years. The Monte Carlo estimate is obtained using 10 000 paths.
is needed however. As shows in Figure 2, there are a few cases where the approximation misbehaves with pricing errors up to 100 basis points.
5
Estimation and Calibration
As mentioned before, the estimation of the parameters proceeds in two steps. In the …rst step, the various parameters depicting the dynamics of the state variables are estimated. In the second step, risk-neutral pricing is achieved through a calibration approach which Q Q generate the risk-neutral parameters Q = Q u; s ; d . Since rt is proxied by the FF rate at weekly frequency, a discrete version of Equation (2) under P is needed: 11
rtn+1 =
tn
+e P
= e
P
tn
t
tn
t
(rtn
rtn + 1
tn )
P
e
+
tn
tn
t
Z
tn+1
tn
+
tn
P
(9) e tn (tn+1 s) dWsP v u u 1 e 2 Ptn t t Zn+1 (equality in law) tn 2 Pt n
=
1
tn
rtn +
tn
tn
+
tn
Zn+1
with weekly observations. It is assumed that
t
(10)
= 1=50
10
.
Estimation of the factor model is performed in two steps via maximum likelihood. First, the transition probability matrices Md , Ms and Mu are estimated using exclusively the t ^d , M ^ s and M^u , the parameters of equation (10) are estimated series. Second, based on M using the series of rt , proxied by the FF rate. The modelization of ( t ; t ) as a joint Markov chain represents a standard regime-switching model readily adapted from [Hamilton (1989)]. Consider the meeting weeks t0 < t1 < t2 < < tn and the corresponding observed target rates f0 ; : : : ; fn . Using this information, the estimation of the transition probabilities emerges from the log-likelihood function: log L =
n X
log Pr
j=1
where fej = (fj ; fj 1 ; : : : ; f0 ), and etj = (
n
tj
= fj etj
tj ; : : : ; t0 ). 11
observable, resorting to the …ltered probabilities log L = n
n X
log
j=1
XX
o
1
= fej
1
o
Since the regime
t
is inherently not
of the regimes generate the function: !
vj ( )M ( ; 0 )M (fj 1 ; fj )
0
et = fej . where vj ( ) = Pr tj = j Hamilton (1989) proposes to compute the vj ’s in a recursive fashion similar to the Kalman …lter where the starting probabilities of the algorithm consist in the stationary probabilities of the current transition matrix M . The short rate parameters are estimated by maximizing the conditional log-likelihood log L =
T X
log f rt ;
t=1
10
t
j r~t 1 ; ~t
1
For the strong solution (3) to hold, which corresponds to Equation (9), it is assumed that t and t remain constant within the time interval t. If it is not the case, the movement in t presumably occurs at the following observation date. 11 The …ltered probabilities provide information about the most likely FOMC regime at every meeting week and can be used to forecast the target rate or to observe the various regime transitions.
12
where r~t = (rt ; rt 1 ; : : : ; r0 ) and ~t = ( t ;
t 1; : : : ; 0)
are based on weekly sampling. This
estimation also resorts to the …ltered probabilities: log L =
T X
log
XX
v t 1 ( )Pt
1
(
t 1;
0
t=1
); ( t ; 0 ) p rt j rt 1 ;
!
t 1;
where v t ( ) = Pr p(y jx; ; ) = p
n
t
1 2
o ret ; et ; ( ) 2 y (x; ; ) exp 2 2
=
2
and (x; ; ) = x(1 )+ . As mentioned before, Pt is represented by equation (1) (the identity matrix) on meeting weeks (outside the meeting weeks). Again, the quantities v t ( )’s are computed through ^, M ^ d, M ^ s and M ^ u. recursive …ltering, while Pt is calculated with the estimated matrices M Risk-neutral pricing is performed by means of calibration where the set of risk-neutral Q Q reversion parameters, Q = Q u ; s ; d , are obtained by minimizing the squared distance between the observed swap rates and their analytical equivalents:
Qt
Q
=
X
S observed (t) i; i
S model t; i; i
Q
2
(11)
:
i
For a de…ned set of reversion parameters, the analytical expression (8) for zero-coupon bond prices is computed. From these prices, the swap rates, S model t; i; i
Q
, are calculated12 . In
spite of the absence of the distributional bene…ts gained by a statistical estimation, the calibration procedure relies on a very simple objective function which substantially reduces the complexity required to determine the appropriate parameters
Q
=
Q u;
Q s;
Q d
: Further-
more, the minimization of the objective function is directly applied on the swap rates, which avoids the inevitable measurement errors generated by curve …tting techniques that compute spot rates or zero-coupon bond prices. To use as much information as possible, the model’s swap rates, S model t; i; i 12
The swap rate S
;
Q
, are build conditionally on the most likely regime bt .
(t), set at time t; satis…es S
;
(t) = P
P (t; T )
i= +1
(Ti
P (t; T ) Ti
1) P
(t; Ti )
where fT +1 ; :::; T g are the payment’s dates. Since the objective function Qt Q is not convex, the minimization routine is initialized using 40 randomly chosen sets of parameters to increase the odds of reaching the global optimum.
13
Table 3: Estimated Transition Matrix for the Three-state Markov 2 0:8969 0:1031 0:0000 2 3 6 (0.0645) 6 pbuu pbus pbud 6 0:0524 0:8994 0:0483 c = 4 pbsu pbss pbsd 5 = 6 M 6 (0.0396) (0.0757) 6 pbdu pbds pbdd 4 0:0000 0:1605 0:8395 (0.1495)
Chain 3 7 7 7 7: 7 7 5
The standard deviation of each probability estimate is in parenthesis. The sample is composed of the target rate, t at each FOMC meeting, providing a sample of 80 observations from the …rst week of August 1998 to the last week of December 2007.
6
Empirical Findings
The estimation of the transition matrix for the FOMC regimes, is reported in Table 3. Almost all transition probability estimates are signi…cant. The probabilities located on the diagonal exihibit high values, from 84% to 90%, a phenomenon consistent with the so-called policy inertia. This suggests that FOMC monetary actions stay in the same regime for long periods. Clearly, short and frequent transitions between regimes are virtually ruled out by the data. As a by-product of the maximum likelihood estimation, the most likely regime’s path using [Viterbi (1967)]’s recursive algorithm is ploted in Figure 313 . This evidence brings further support the view where FOMC monetary actions are consistent with a three-state hidden Markov chain. The …rst panel of Table 4 contains the transition probabilities of the target rate in each FOMC regime. Some estimates are not signi…cant perhaps as a consequence of the reduced number of occurrences for some states of the Markov chain. It may come as no surprise that the target rate exihibits an inertia behavior in the stable regime with a probability of 94%. Consistently, the target rate shows an improbable stable behavior in both the upward and downward regimes. An interesting contrast emerges when comparing these two regimes.
In the upward regime, there exists a probability of 86% (4%) of observing the
target rate increasing by 25 bps (50 bps). But in the downward regime, the target rate shows a probability of 24% (57%) of declining by 25 bps (50 bps). Thus, it seems that the FR acts di¤erently whether it undertakes an expansionary or restrictive monetary policy and behaves more prudently in latter case. The second and the third panels report the estimates 13
The most plausible regime’s trajectories, etn = 1 ; :::; tn , maximizes the conditional joint mass function P etn etn . It may di¤er from the regime that maximizes the …ltered probabilities P t et , t 2 ft1 ; :::; tn g . Indeed, in the latter case, the information up to time t is used while in the former case, the entire time series of target rates is used in addition to the dynamics of the Markov chain.
14
Figure 3: Most Likely Regime’s Path
.
Table 4: Estimates of the Target Rate Transition Matrices and of the Short Rate TargetReversion and Volatility Parameters
ParametersnRegime ( )
Downward
Target rate, t Stable
Upward
Panel 1: Target rate’s transition probabilities p p p p p
( 50) ( 25) (0) (25) (50)
P
0.5745 (0.1432) 0.2378 (0.1164) 0.1877 -
0.0215 (0.0397) 0.0271 (0.0349) 0.9403 0.0111 (0.0262) 0.0000
0.1038 0.8580 (0.0803) 0.0382 (0.0377)
Short rate, rt Panel 2: Discrete time parameters 0.561717 (0.116831) 1.0 (NaN) 0.00236480 (0.00015928) 0.00038581 (0.00002119)
0.623503 (0.086538) 0.00170902 (0.00009238)
Panel 3: Continuous time equivalent parameters 41.245 230. 26 0.023895 0.0082798
48.842 0.018233
Panel 1 reports the joint estimates of the transition probabilities (standard deviation) of the target rate across the three FOMC regimes. Panel 2 reports the target’s reversion and volatility estimates of Equation (10) across the three FOMC policy regimes. Panel 3 reports the continuous time equivalent parameters. 1 P Transformations, using equation (10), are applied to the point estimates: P = = t ln 1 q 2 1 2 t ln 1 =1 1 : However, for the case where ! 1, it implies that ! 1. Thus, s and s are computed based on the assumption that s = 0:99:
15
Table 5: Summary Results From the Calibration Procedure for the Risk-Neutral Parameters, Q Q Q = Q u ; s ; d , and for the Swap Rate Pricing Errors Q u
Mean Median Standard deviation Min Max
Q s
Q d
7,05 3,07 0,84 6,40 2,68 0,01 4,84 4,20 2,90 -0,67 -1,48 -3,56 25,66 13,13 8,71
RMSE(bps) 8,83 3,19 9,75 0,10 35,16
This table reports descriptive statistics based on the 38 quarters over which the calibration procedure was applied, using Equation (11). For each quarter, a root mean square error (expressed in basis points) based on the 2, 5, 7 and 10 years swap rates is computed.
of the spot rate process parameters. They are signi…cant in all three regimes. It is worth noticing that in the stable regime, the short rate ‡uctuates as a random walk since s = 1. In the other regimes, the large the target rate. parameters,
P
re‡ect the strong reversion of the FF spot rate toward
This is consistent with the evidence showed in Figure 1. The volatility
are also signi…cant14 .
The …ndings from the sequential calibration based on equation (11) performed at a quarterly frequency, are reported in Table 5. It shows descriptive statistics for the targetreversion parameters estimated under the three FOMC policy regimes and, on the last column, the RMSE swap rate errors. The median (mean) amounts to 3.19 (8.83) basis points which is highly competitive with the …ndings found in the literature (see, for instance, [Piazzessi (2005)]). Globally, the model explains well the attributes of the swap curves although the standard deviation stands at 9.75 basis points. Although not reported for space limitations, this standard deviation is probably driven by the three quarters where the model underperformed with pricing errors in the vicinity of 30 basis points. Risk-neutral reversion parameters appear reasonable at median values of 6.40, 2.68 and 0.01 under the FOMC upward, stable and downward regimes, respectively. Figure 4 plots the risk-neutral reversion parameters across the 38 quarters. As indicated in Table 5, there are several quarters where bQ d becomes negative but of small magnitude. On the other hand,
the parameters demonstrate some instability, particularly in the case of bQ u with a surprising
spike at quarter 2001
III. This unappealing feature of the model might result from the
nature of the calibration procedure that utilizes cross-sectional information embodied in the swap curve, in a sequential fashion. 14
Since lim !1 = 1, the mean reverting parameter Ps in the stable regime is arti…cially in‡ated. However, it doesn’t impact the pricing formula which use the risk-neutral parameter.
16
Figure 4: Evolution of The Calibrated Parameters,
Q
=
Q u;
Q s;
Q d
, across the 38 quarters
The three parameters are calibrated from the 2, 5, 7 and 10 years swap rates available on the last Friday of each quarters.
Figure 5 shows the sensitivity of the swap curve with respect to movements in the target rate. Each graph plots a collection of swap curves for maturities reaching 10 years for each of the FOMC policy regimes. These curves are generated by the model when the target rate ranges from 0:5 % to 7 % by steps of 25 basis points and when the parameters, Q
=
Q u;
Q s;
Q d
, are set to their average value (…rst line of Table 5). An informal visual
inspection of swap curves contained in the sample used in this study, indicates that there are pretty much captured by those ploted in Figure 5. The model re‡ects well some well known stylized facts since short term swap rates display more volatility than their long term counterpart. Also, swap curve movements tend to occur in a parallel fashion. The model reveals that non-linear movements in the swap rates are likely to take place only at the short end of the curve in an asymetrical fashion that depends on the monetary policy regime. For instance, when monetary authorities undertake a "stable" monetary policy, the model tells us that assymetrical movements in the short term swap rates occur only in regions where the target rate level is either very high or very low. Other non-linear movements occur in the upward (downward) regime when the target rate is at a very high (low) level. More interestingly however, Figure 5 points out that as long the FR implements monetary actions that fall within the same regime, the slope of the swap yield curve does not exhibit major change for any target rate movement of 25 or 50 basis points. This …nding seems in line with the policy intertia view because slope changes occur only when there is a regime
17
Figure 5: Swap Curve Sensitivities to the Initial FOMC regime and to the Target Rate Level
Upward initial regime
Stable initial regime
0.08
0.08
0.07
0.07
0.06
0.06
Downward initial regime 0.07
0.06
0.05
0.05
0.05
Swap rates
Swap rates
Swap rates
0.04 0.04
0.04
0.03 0.03
0.03
0.02
0.02
0.01
0.01
0.02
0
0
2
4 6 8 Time to maturity (years)
10
0
0.01
0
2
4 6 8 Time to maturity (years)
10
0
0
2
4 6 8 Time to maturity (years)
10
Each of the three Figures (one for each regime) shows the variation of the swap curve to the initial target rate, Q Q going bottom up from 0 = 0:5% to 0 = 7% by step of 25 basis points. The parameters Q = Q u; s ; d have been set to their average values while the other parameters are set to their point estimates.
transition. For instance, a decrease in the target rate, say from 2.50% to 2% within the stable regime induces a parallel downward shift to the swap curve. However, the same change accompanied by monetary actions transiting from the "stable" to the "downward" regime generate a drastic change to the shape of yield curve which becomes inverted with a humped.
7
Conclusion
This study proposes a no-arbitrage interest rate model whose distinctive features include regime-switch combined with observability of the state variables. The design of the factor model naturally introduces the concept of FOMC policy regimes whose dynamics are proxied by a three-state hidden Markov chain. On that …rst conceptual layer, the joint modelization of the target and Fed fund rates is added. The target is assumed to behave as discrete Markov chain with transition probabilities conditioned upon the current FOMC regime. In a fashion consistent with the literature, the Fed Fund rate (which acts as a proxy for the 18
short rate) is driven by a target-reverting process with regime-switching parameters. Empirical …ndings demonstrate that the parametrization of the target rate and its FOMC regimes as a joint Markov chain is supported by the data and generation of the most likely trajectory of the target rate appears remarkably close to its true path. Good results are also obtained for the Fed Funds rate process since the reversion and volatility parameters are signi…cant in all regimes. Important reversions towards the target are observed under the upward and downward regimes. Such is not the case in the stable FOMC regime where the Fed Fund rate displays a random walk behavior. As expected, the volatility of that rate is not stationary and ‡uctuates according the to FOMC regimes. However, the introduction of a joint Markov chain into the risk-neutral pricing framework is not costless because the exact solution for the zero-coupon bond prices is impractical in this case. To bypass this obstacle, an approximate recursive solution is generated. Numerical simulations reveal that this approximation performs well for a vast variety of risk-neutral target-reversion parameters. Parameter extraction based on swap rates is achieved via a sequential calibration approach. Overall, the three factor model …ts the swap curve in a very competitive manner. Risk-neutral target-reversion parameters observed in the upward and downward FOMC regimes show reasonable values while their stable-regime counterparts have a tendency to become negative. Finally, a sensitivity analysis revealed that target rate movements generally trigger parallel movements in the swap curve while transitions of di¤erent FOMC regimes can drastically change its shape.
19
Appendix A
Proof of Theorem 6
The proof is based on an induction argument on the number, m, of time periods between t and T . If t = tn and T = tn+1 , then, conditionnally upon Gtn , the SDE (3) admits the strong
solution
ru =
tn
R tn+1
Therefore,
tn
Q tn (u
+e
tn )
(rtn
tn ) +
tn
Z
u
Q tn (u
e
s)
dWsQ ; tn
t
tn+1
tn
ru du obeys to a Gaussian distribution and the conditional value of the zero-
coupon bond is given as in Vasicek (1977): P t; T ; rtn ;
tn ;
tn
tn ; kn
tn ;
rtn + B1
tn ; kn
= exp A1
The coe¢ cients A1 ( ; k) and B1 ( ; ; k) given by equation (7) converge to zero as As induction hypothesis, assume that the result (6) holds when T
t = (m
1)
t
! 0: t.
Let t = tn and T = tn+m . Note that kn+1 = kn + 1 if kn < k and 0 otherwise. Thus P tn ; tn+m ; rtn ; tn ; tn ; kn Z tn+m Q = E exp ru du Gtn tn " " Z Q
= E
Q
= E
tn+1
exp
ru du E
Z
exp
=
;
Q
E h
exp
exp
ru du exp Am
tn+1
1
Gtn+1
#
Gtn
tn+1 ; tn+1 ; kn+1
tn+1 ; kn+1
rtn+1 + Bm
# Gtn
1
tn+1 ; tn+1 ; kn+1
Gtn
tn+1
ru du exp Am
tn
= ;
ru du
ru du P tn+1 ; tn+m ; rtn+1 ;
Z
!
tn+m
tn+1
tn+1
tn
Q
Z
tn
tn Z tn+1
= EQ exp X
Q
tn+1
=
tn ;
tn
1(
; kn+1 ) rtn+1 + Bm
i
1(
; ; kn+1 ) Gtn
where the last equality arises from assumption 1. Rt Given Gtn ; the conditional distribution of tnn+1 ru du and rtn+1 is bivariate Gaussian. Therefore, Z tn+1 Q E exp ru du + Am 1 ( ; k) rtn+1 Gtn tn 1 0 h R i tn+1 EQ r du + A ( ; k) r G u m 1 tn+1 tn h tnR i A = exp @ t n+1 Q 1 ru du + Am 1 ( ; k) rtn+1 Gtn + 2 Var tn 20
where E
Q
Z
tn+1
ru du Gtn
tn
vR
tn
= VarQ
Z
tn
=
Q
t
Q
tn
EQ rtn+1 Gtn
tn
Q tn
= e
t
rtn +
tn
ru du Gtn
tn
=
t
2
Q
2
tn
Q tn
tn
!
tn ;
tn ;
2
tn+1
tn
rtn +
tn
1 + 2
tn
Q tn
!
;
2
vr cr;R
tn
= Cov
Q
= VarQ rtn+1 Gtn
tn
Z
tn
=
2
tn
;
2
tn+1
tn
tn
Q
ru du; rtn+1 Gtn
tn
=
tn
2
Q
tn
:
2
tn
i i hR hR t t Note that whenever tn+1 tn = t is small, then EQ tnn+1 ru du Gtn , VarQ tnn+1 ru du Gtn , i hR t VarQ rtn+1 Gtn and CovQ tnn+1 ru du; rtn+1 Gtn are in the neighborhood of zero and EQ rtn+1 Gt = rtn , justifying the use of the approximation exp (x) = 1 + x. Since the model assumes that the transitions of regimes are not a¤ected by the level of the target rate, one may de…ne X Am 1 tn ; kn = Am 1 ( ; kn ) Q tn+1 = ; tn+1 = tn ; tn ;
= = Am Bm
1
1
tn ; kn
=
tn ;
tn ; kn
=
X
X
X
X
Am
1
( ; kn ) Q
tn+1
=
tn
Am
1
( ; kn ) Q
tn+1
=
tn
;
A2m
1
( ; kn ) Q
tn+1
=
tn
;
Bm
1
( ; ; kn ) Q
tn+1
;
21
= ;
tn+1
X
=
Q
tn+1
tn ; tn
=
:
tn ; tn
!!
Therefore, getting back to the bond price,
=
X ;
Q
= 1+
B B exp B @ h
tn+1
0
XB B @
i ru du Gtn + Am 1 ( ; kn+1 ) EQ rtn+1 Gtn i ru du Gtn + 12 A2m 1 ( ; kn+1 ) VarQ rtn+1 Gtn i hR t Am 1 ( ; kn+1 ) CovQ tnn+1 ru du; rtn+1 Gtn + Bm 1 ( ; ; kn+1 ) i = ; tn+1 = tn ; tn
Q
=
h
0
1+@
tn+1
EQ
tn Q tn
= ;
+ 12 vR
= exp Am
rtn +
+ 12 vR tn+1
tn Q tn
1 + Am
hR
tn+1 h Rtn t + 12 VarQ tnn+1
;
=
tn ; tn ; kn
P tn ; tn+m ; rtn ; 0
tn
tn ; kn tn ; kn
+
+
tn
1
1 2A
rtn + Bm
tn
1 2 2 Am 1
tn ; tn
=
+ Am
tn Q tn
t
i
tn ; kn+1 tn ; kn+1
+ Am
( ; kn+1 ) vr +Bm 1 tn ;
e vr
Q tn
tn
t
( ; kn+1 ) e
tn
Am
Am
tn ; tn ; kn
1
Q tn
t
rtn +
( ; kn+1 ) cr;R
C C C A
(12)
tn
tn
tn
tn ; kn+1
rtn + Am
tn ; tn ; kn
rtn + Bm
1
1
1
1
tn ; kn+1
tn ; kn+1
cr;R
tn
tn
+
+ Bm
tn Q tn
1
1 C C A
t
tn
tn ; tn ; kn+1
1 A (13)
The two approximations are based on the Taylor expansion of exp (x) for x in the neighborhood of zero. Indeed, one can show from a recursive argument that Am ( ; k) ! 0 and Bm ( ; k) ! 0 as t ! 0.
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24
Technical Report This is not part of the paper
B
Estimation of the regime’s most likely path
Let et = ( 1 ; 2 ; :::; t ) and et = ( 1 ; :::; t ) : The most plausible regime’s trajectories eT maximizes the conditional mass function m (e xT je yT ) = P eT = x eT eT = yeT : Since P eT = x eT eT = yeT P(eT =e xT ;eT =e yT ) xT ; yeT ) = P eT = x eT ; eT = yeT : ; then eT also maximises the joint mass function q (e yT ) P(eT =e Let e i (t) = max q et 1 ; t = i; t et
and note that j
(t + 1) = max q et ; et
= max P et
1 ;i
t+1
t+1
1
= j; et+1
=jj
t
= i P(
t+1
j
= i;
t
= max P
t+1
=jj
t
= i P(
t+1
j
t
= i;
= max P
t+1
=jj
t
= i P(
t+1
j
t
= i;
= max
(
i i i
i;j pi
t+1
t) i
et 1 ;
t)q
t ) max q et 1
t) i
(t)
t
= i; et
et 1 ;
t
= i; et
(t) :
This leads to the Viterbi algorithm: Initialization i i
where
(1) = max q ( et
1
= i;
1)
=
(i)
1
(1) = 0:
( ) denotes the stationnary distribution of the Markov chain of the regime.
Iteration
and
j
(t + 1) = max
j
(t + 1) = arg max
i
i;j pi i
(
t) i
t+1
i;j pi
(
t+1
i
(T )
t) i
Final step max q (e xT ; yeT ) = max
x eT ;e yT
i
and
T
= arg max i
Reconstruction t
=
t+1
25
(t + 1)
i
(t)
(T ) :
(t) :
=